The Monotone convergence theorem:
If $f_i: X\rightarrow[0, \infty]$ are $\mu$-measurable functions such that $f_i\leq f_{i+1}$ then $$ \int_X \lim_{i}f_id\mu = \lim_{i}\int_X f_id\mu $$
We know it holds for the Lebesgue measure $\mu$. I want to ask are there any requirements on the measure $\mu$? Are there any other example measures that it doesn't hold? On what conditions of $\mu$, it can/can't hold?
There aren't. Fatou, MCT and DCT hold for all $\sigma$-additive measures. See for instance the chapters on measure theory in Royden's Real Analysis, or any book that treats measure theory.